Dipole in a uniform field

Problem

Conceptual: dipole in a uniform field. An electric dipole (two charges $+q$ and $-q$ at distance $d$) is placed in a uniform electric field. Does it experience a net force, a torque, or both? Explain in five lines. And in a non-uniform field, what changes?

Solution

Net force in a uniform field. Forces $\vec F_+ = +q\vec E$ and $\vec F_- = -q\vec E$ act on the two charges. In a uniform field $\vec E$ is the same at both points, so the two forces are equal in magnitude and opposite in direction: $\vec F_\text{net} = q\vec E + (-q)\vec E = \ev{\vec 0}.$ The dipole’s centre of mass does not accelerate: no translation.

Torque. The two forces, although balanced, are applied at different points: they form a couple that tends to rotate the dipole. The torque is $\ev{\vec\tau = \vec p \times \vec E}, \qquad \tau = pE\sin\theta,$ where $\vec p = q\vec d$ is the dipole moment and $\theta$ the angle between $\vec p$ and $\vec E$. The torque vanishes when $\vec p$ is aligned with $\vec E$ ($\theta = 0$): this is the configuration of minimum energy and stable equilibrium, towards which the dipole tends to orient itself. It is the mechanism by which polar molecules (such as water) align with an applied field.

Non-uniform field. If $\vec E$ varies from point to point, the forces on the two charges no longer cancel exactly: alongside the torque a net force also appears, directed towards the region of stronger field (for an aligned dipole). This is why an electrified comb attracts bits of neutral paper: the comb’s non-uniform field induces, and then drags, the dipoles.

Links

Topics: Campo elettrico e potenziale Concepts: Dipolo elettrico · Campo elettrico Skills: Principio di sovrapposizione Methods: Metodo del gradiente Objects: Sfera carica